TITLE: How to prove that $(\mathbb Q,+)$ is not isomorphic to $(\mathbb Q^*,\times)$?
QUESTION [5 upvotes]: Given that there is the additive group $\mathbb Q$ of rational numbers, and the multiplicative group $\mathbb Q^*$ of non-zero rational numbers, prove that $(\mathbb Q,+)$ is not isomorphic to $(\mathbb Q^*,\times)$.

How many methods can you think of and can you provide a complete solution?
I am a self-learner of maths and feel difficult to offer a rigorous proof, but here are my thoughts:

I could try to assume a isomorphism $\theta$ exists between the two groups and prove that $\theta$ cannot exist.
I could try to find some property which should preserve under isomorphism but is satisfied only by one of the groups.

However I could not proceed in either direction, could someone please help?

REPLY [9 votes]: Check for the existence of elements of order $2$.
In the light of this your next question is probably about $(\Bbb Q^*_{>0},\times)$. Then we can resort to the fact that $(\Bbb Q,+)$ is divisible.